I am trying to construct a model for use in a video game

Basically I have boiled things down to the following situation that needs to

be modelled:

Think of a snooker/pool table (without the pockets) with two balls on it.

The balls bounce around colliding with the sides of the table and each

other. Perfectly elastically for simplicity.

For further simplicity, there is no friction. Maybe they have unit mass.

Basically picture two circles bouncing around inside a rectangle.

My question is:

Given:

1) the dimensions of the rectangle and the circles.

2) the initial velocities of the circles

Is there an equation where I can find the location of the circles at time t?

e.g.

circle_one_centre(x,y) = f(t)

circle_two_centre(x,y) = f(t)

If so, how do i go about finding this equation?

what about the same thing for n circles?

Different shaped tables??

Any ideas?

Thanks for any help

Dean

I hope you reconsider your strategy.

If you simplify away all the effects that make the game interesting, you

will have little left.

- Matt

If you simplify away all the effects that make the game interesting, you

will have little left.

- Matt

In article <fX4%d.3809$ XXXX@XXXXX.COM >, XXXX@XXXXX.COM

says...

No. Even this simple system is chaotic, and there is no finite closed

form equation that will give the result you want. You have to calculate

to the next collision, then the one after that, and so on.

You add the problem of multi-ball collisions.

Less of a problem - I don't think this makes the situation all that much

more difficult.

- Gerry Quinn

says...

No. Even this simple system is chaotic, and there is no finite closed

form equation that will give the result you want. You have to calculate

to the next collision, then the one after that, and so on.

You add the problem of multi-ball collisions.

Less of a problem - I don't think this makes the situation all that much

more difficult.

- Gerry Quinn

I once read a paper on Gamasutra.com, where circle-circle collision

detection/handling was discussed.

Perhaps something you could use, just register there and search

for "Pool Hall Lessons".

No friction is handled there, though.

hth, Markus

>> > Can you get a finite closed form equation for the ball position?

From what I can gather the term means (from the posts and google), I was

indeed looking for a closed form equation.

Also, I know you had to repeat yourself Gerry, but I wasn't sure that

chaotic systems cannot be defined by closed form equations. I was going to

quote the fox-rabbit population equation, but then I remembered that that

was iterative too. But, even if chaotic systems cannot be defined by closed

form equations, you did not show/"prove" that the system under discussion

was indeed chaotic! (which was implicit in "if not, why not" question)

Anyway, please read my new post "It has failed, how do I handle my

collisions now?"

This shows what I am really after.

From what I can gather the term means (from the posts and google), I was

indeed looking for a closed form equation.

Also, I know you had to repeat yourself Gerry, but I wasn't sure that

chaotic systems cannot be defined by closed form equations. I was going to

quote the fox-rabbit population equation, but then I remembered that that

was iterative too. But, even if chaotic systems cannot be defined by closed

form equations, you did not show/"prove" that the system under discussion

was indeed chaotic! (which was implicit in "if not, why not" question)

Anyway, please read my new post "It has failed, how do I handle my

collisions now?"

This shows what I am really after.

In article <vTK0e.1868$% XXXX@XXXXX.COM >, XXXX@XXXXX.COM

says...

It's not my job to prove correct answers, and I'm not a mathematician.

But in a nutshell:

The system is clearly chaotic as a result of its physical configuration.

The ball positions at time t after a collision depend sensitively on the

exact trajectory of the balls before the collision. Vary the initial

position of the collision by one millimetre, and a moment later the

position will be a metre from where it was. This kind of process

happens repeatedly, and the ball position at later times as a function

of the initial trajectory becomes scattered all over the table (it may

be more dense in certain parts of the table), even for initial

trajectories that were very close together In other words, chaos in a

typical form.

As t goes to infinity, the functions defining the position at t have to

come arbitrarily close to a non-differentiable curve. Any closed form

analytic solution will run towards infinitely many terms. (In practise,

the functions would be horrific even after two or three collisions,

though in principle it might be possible to find functions that will

work through a few iterations.)

For your very simple system, calculate the next collision position,

calculate the results of the collision, and repeat ad infinitum. For

balls with real physics, I recommend some intensive googling.

- Gerry Quinn

says...

It's not my job to prove correct answers, and I'm not a mathematician.

But in a nutshell:

The system is clearly chaotic as a result of its physical configuration.

The ball positions at time t after a collision depend sensitively on the

exact trajectory of the balls before the collision. Vary the initial

position of the collision by one millimetre, and a moment later the

position will be a metre from where it was. This kind of process

happens repeatedly, and the ball position at later times as a function

of the initial trajectory becomes scattered all over the table (it may

be more dense in certain parts of the table), even for initial

trajectories that were very close together In other words, chaos in a

typical form.

As t goes to infinity, the functions defining the position at t have to

come arbitrarily close to a non-differentiable curve. Any closed form

analytic solution will run towards infinitely many terms. (In practise,

the functions would be horrific even after two or three collisions,

though in principle it might be possible to find functions that will

work through a few iterations.)

For your very simple system, calculate the next collision position,

calculate the results of the collision, and repeat ad infinitum. For

balls with real physics, I recommend some intensive googling.

- Gerry Quinn

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